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I am a beginner in representation theory. I have some questions about Cartan involution. The following is the link in wikipedia

http://en.wikipedia.org/wiki/Cartan_involution

My question is about $\mathfrak{su}(n)$. If $\mathfrak{su}(n)$ is the Lie algebra of compact semisimple group, the only Cartan involution is $\theta_0:X\mapsto X$. So why $\theta_1$, $\theta_2$ and $\theta_3$ are there? In what situation are they available?

Qinghua Pi
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These maps $\theta_1,\theta_2,\theta_3$ are involutions in the sense that they are self-inverse Lie algebra automorphisms of $\mathfrak{su}(n)$. They aren't Cartan involutions because the associated bilinear form $B_\theta$ isn't positive definite.

bradhd
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  • So, they are examples of involutions of $\mathfrak{su}(n)$ which are not Cartan involutions. I understand, thanks a lot ! – Qinghua Pi Jan 20 '14 at 16:43